Proprioceptive Actuator Design in the MIT Cheetah: Impact Mitigation and High-BandwidthPhysical Interaction for Dynamic Legged Robots

Patrick M. Wensing1, Albert Wang1, Sangok Seok2, David Otten3, Jeffrey Lang3, and Sangbae Kim1

Abstract—Designing an actuator system for highly-dynamiclegged robots has been one of the grand challenges in roboticsresearch. Conventional actuators for manufacturing applicationshave difficulty satisfying design requirements for high-speedlocomotion, such as the need for high torque density and theability to manage dynamic physical interactions. To addressthis challenge, this paper suggests a proprioceptive actuationparadigm which enables highly-dynamic performance in leggedmachines. Proprioceptive actuation uses collocated force controlat the joints to effectively control contact interactions at thefeet under dynamic conditions. Modal analysis of a reducedleg model and dimensional analysis of DC motors address themain principles for implementation of this paradigm. In therealm of legged machines, this paradigm provides a uniquecombination of high torque density, high-bandwidth force control,and the ability to mitigate impacts through backdrivability. Weintroduce a new metric named the ‘impact mitigation factor’(IMF) to quantify backdrivability at impact, which enables designcomparison across a wide class of robots. The MIT Cheetah legis presented, and is shown to have an IMF that is comparableto other quadrupeds with series springs to handle impact. Thedesign enables the Cheetah to control contact forces duringdynamic bounding, with contact times down to 85 ms and peakforces over 450 N. The unique capabilities of the MIT Cheetah,achieving impact-robust force-controlled operation in high-speed3D running and jumping, suggest wider implementation of thisholistic actuation approach.

I. INTRODUCTIONPhysical interactions with the environment play a crucial

role in many emerging applications of robotics. Whether inlegged locomotion or disaster response, the need to controland exploit interaction with the environment introduces uniquechallenges to actuator design. As we focus on these applica-tions, it is imperative to shift our actuator paradigms awayfrom technologies that have been optimized for the assemblyline floor. Most manufacturing robots have been designed toperform rapid and accurate pick-and-place position controlsequences in well-known environments. Future mobile robots,however, must be capable to manage broader dynamic andforce-critical interactions in unstructured environments. Theactuator paradigm presented in this paper offers an approachfor electromagnetic (EM) actuators to address these needs.

The design of actuators to manage physical interactionsis particularly challenging for application to dynamic leggedlocomotion. Legged locomotion involves repeating dynamicevents such as impact, high-force interaction with uncertainterrain, and rapid leg swing. These events place unique and

Authors are with the 1Department of Mechanical Engineering at theMassachusetts Institute of Technology, Cambridge, MA, 02139, USA, 2NaverCorporation, Seongnam, South Korea, and the 3Department of ElectricalEngineering and Computer Science at the Massachusetts Institute of Tech-nology, Cambridge, MA, 02139, USA. corresponding email: sangbae atmit.edu

Geared Motor with Torque(Force) Sensor

Series Elastic Actuator

Leg Leg

Spring, Encoder

motor

High Gear Ratio Transmission Stiff

Sensor

motor

High Gear Ratio Transmission

High torque density motor

Low Gear Ratio Transmission

Low inertia leg

Proprioceptive actuator

(a) (b)

(c)

Geared Motor with Torque(Force) Sensor

Series Elastic Actuator

Leg Leg

Spring, Encoder

motor

High Gear Ratio Transmission Stiff

Sensor

motor

High Gear Ratio Transmission

High torque density motor

Low Gear Ratio Transmission

Low inertia leg

Proprioceptive actuator

(a) (b)

(c)

Geared Motor with Torque(Force) Sensor

Series Elastic Actuator

Leg Leg

Spring, Encoder

motor

High Gear Ratio Transmission Stiff

Sensor

motor

High Gear Ratio Transmission

High torque density motor

Low Gear Ratio Transmission

Low inertia leg

Proprioceptive actuator

(a) (b)

(c)

Propriocep(ve**Actuator*

(c)%%

(a)%% (b)%%

Geared*Motor*with*Force/Torque*Sensor* Series*Elas(c*Actuator*

Fig. 1. Three different electromagnetic (EM) actuator concepts. (a) High-ratiogeared motor with torque sensor, (b) Series elastic actuator, (c) Proprioceptiveforce control actuator.

often conflicting requirements on the design of EM actua-tors. Design processes for EM components are traditionallygoverned by performance metrics such as the motor con-stant, torque constant, peak torque, continuous torque, thermalcapacity, and others [1]. Beyond the EM components, thetransmission (speed reducer) is a critical additional componentwhich affects the control bandwidth and ability to handlecollisions with the environment [2]. Multi-objective designtrade offs thus need to be weighed in the selections of boththe EM components and transmissions. Three common EMactuator concepts that manage these trade offs for robotics areshown in Fig. 1 and are discussed throughout the paper.

We propose to address this intricate actuator design problemthrough a new paradigm for EM actuators in legged machinescalled proprioceptive actuation. As a key feature, this allows‘proprioceptive’ foot force control through internal torquecontrol. This can be accomplished without exteroceptive sen-sory feedback that is known to cause non-collocated sensingproblems upon collision which result in contact instability [3].The paradigm of proprioceptive actuation has had successin low-force haptic devices such as the PHANTOM [4], butpreviously has not been employed for high-force applicationssuch as legged locomotion. Overall, the paradigm eliminatesthe need for physical springs, stiffness modulating mecha-nisms, and force/torque sensors, and enables high-bandwidthforce control without contact force feedback.

Proprioceptive actuation is a unique paradigm to addressmany of the design challenges facing legged locomotion.These challenges are presented in Section II with a summary

of the previous approaches developed to handle them. Throughmodal analysis of a simple prototype of a legged system,and dimensional analysis of motor performance across scale,Section III shows that the proprioceptive paradigm is uniquelysituated to manage high-force, high-bandwidth contact interac-tions imperative to high-speed locomotion. We view handlingimpact as the most difficult of these contact interactions,and propose a new metric in Section III-B called the ImpactMitigation Factor (IMF). The IMF quantifies the normalizedinertial impedance of a floating-body robot, capturing theeffects of actuator design to reduce impulsive forces at impact.

The MIT Cheetah design, presented in Section IV, embodiesthe principles of proprioceptive actuation. By minimizing theoverall impedance of the actuator, the design is able to mitigateimpacts comparably to existing quadrupeds with series elasticactuators (SEAs), while maintaining high-bandwidth open-loop force control (Section V). A discussion in Section VI de-tails the high-level benefits on this paradigm which embracesa notion of ‘less is more’ in mechanical design. We believethis approach provides a robust and practical new paradigmfor legged machines.

Portions of this article (in Section III-C) have appearedpreviously in a conference paper [5]. Significant additionaltheoretical analysis and experiments are reported here beyondthe previous work.

II. CHALLENGES IN ACTUATOR DESIGN FOR HIGH-SPEEDLEGGED ROBOTS

One of the main challenges in actuator design is to identifycritical metrics for the target application and tradeoffs betweenthe metrics across design parameters. For dynamic leggedlocomotion, we identified the following key actuator char-acteristics: 1) high mass-specific torque (torque density), 2)efficiency, and 3) impact mitigation capability. The followingsubsections describe these three aspects in more detail.

A. Torque density

Achieving high torque with a minimum actuator mass iscritical in running robots. This is due to in part to the highground reaction forces (GRFs) required to propel the body.The maximum normal GRF on each leg is about three timesthe bodyweight in a human running at 4.5 m/s [6]. In data for adog galloping at 9 m/s [7], it reaches 2.6 times the bodyweightduring a ground phase of approximately 70 ms (20% of gaitperiod). While long legs provide a large workspace, theyintensify the torque requirements to produce these forces.To meet GRF requirements at the foot, large moment armsdictate high joint torques. For example, the maximum torqueat the knee required for a 75 kg human to jump is around200 Nm [8]. This is remarkable considering that the torquefrom a 1270 kg Toyota Corolla is 170 Nm at 4000 rpm [9].

Simply employing large motors to meet torque requirementsis not viable in mobile robot design. Higher actuator massescontribute to GRF requirements, and thus increase torquerequirements. This relationship highlights the importance ofmaximizing torque density for actuators in legged machines.

The power of the motor often becomes a major metric indesigning motors for traditional applications. However, thehigh power achieved by high-speed continuous operation doesnot apply to legged machines, where operation is mostlyintermittent and varying directions continuously. In fact, thepower density of EM actuators themselves (continuous upto 7 kW/kg [10], 3-5 kW/kg [11]) can significantly exceedthat of biological muscle (Max. 0.3 kW/kg) [12], and ofmost manufacturer-recommended operating specifications forcommercial motors.

B. Energy Efficiency

Energy efficiency is a critical metric for legged machines.Locomotion is a typically energy dissipative process and theloss mechanisms are categorized into three major modes:Joule heating, transmission, and interaction losses [13]. Wepreviously described design principles to minimize these lossesfor highly-efficient legged robots. Their implementation, in thedesign of the MIT Cheetah, has led to a machine that rivalsthe efficiency of animals [13]. However, this design was notexplicitly optimized for efficiency.

Transmission selections in particular have coupled implica-tions for Joule heating and interaction losses. Low gear ratioswill reduce the reflected inertia but can cost higher energyin generating torque via Joule heating. On the other hand, ifthe gear ratios used in a robot are too high for its application,they may improve energetics in theory but can break a leg uponcontact or even could prevent a desired dynamic motion due toexcessive mechanical impedance. This tradeoff should furtherbe considered based on the robot’s functional requirementssuch as payload, required travel distance per charge, andtravel speed. For instance, a load-carrying robot walking atslow speeds could energetically benefit from a higher gearratio in comparison to an agile robot running at high speeds.Gear efficiencies and frictional properties play a large rolein balancing energetic tradeoffs. Yet, while there exist manymethods to identify gear energetics of a fabricated design[14], accurately modeling gear energetics a priori for designoptimization is complicated by a variety of factors [15]. Still,whereas energy efficiency is a metric we should seek tomaximize, impact mitigation is an essential requirement.

C. Impact mitigation

While satisfying the torque and power requirements, the legshould be able to mitigate impact forces upon collision. Thereare a number of approaches in the literature which may useactive control or passive mechanical dynamics to minimizecollisional forces surrounding an impact event.

Researchers have developed several approaches that employimpedance control with high-gear-ratio actuators to achieveforce reduction after impact. Apparent impedance can bemodulated by implementing impedance control [16] withtorque sensors at each joint [17], [18]. However, this controlapproach used with high-gear-ratio actuators has not demon-strated robustness to impacts or high-bandwidth force controlcapabilities for dynamic legged locomotion. Previous workhas addressed human-robot impacts in industrial settings with

(a)

m`

ki

mb

✓

f

yI

(b)

m`

ki

I

mb

ks

f

y

Fig. 2. Prototype systems to understand the effects of actuation and legdesign on mechanical bandwidth and impact force magnitude. (a) Prototypewith rigid transmission and (b) prototype with series elastic actuation (SEA).The SEA rotational actuator is assumed connected to mass b, with the SEAspring between the actuator output and the pinion.

the DLR Lightweight Robot III [19]. Initial impact eventsare detected, and effective control post impact is shownto reduce overall collision severity. As a main difference,proprioceptive actuation provides a mechanical approach tominimize the severity of initial and post-impact forces, whichcan otherwise be detrimental in high-speed collisions withoutsufficient compliance.

The series elastic actuation (SEA) paradigm [20], [21],[22] has been employed in legged machines to reduce ini-tial rigid-body impacts by purposefully adding mechanicalelasticity in series with an actuator. More recently, numerousdesigns have been presented for variable stiffness actuators(VSAs) to mitigate impacts. Stiffness modulation in VSAshas been achieved through a variety of mechanisms. Designsmay strategically load nonlinear springs [23] to modulatestiffness or may use linear springs by stretching them ina nonlinear way [24]. Vanderborght et al. [25] provide athorough review and classification of the many ways to provideand modulate stiffness in VSAs. Other SEA designs, suchas in the quadruped StarlETH, have demonstrated successfulexecution of controlled variable leg impedance using fixedstiffness joint SEAs [26]. While SEAs offer a great potentialactuator solution for legged robots, their force bandwidth cansuffer in comparison to designs without added compliance.The next section will quantify this effect in a simple leg model.

III. PROPRIOCEPTIVE FORCE CONTROL ACTUATION

A. Impact force analysis in a simplified leg model

This section studies how the design parameters of anactuation system affect its impact dynamics and force control.A simple model, shown in Fig. 2, was developed to captureessential design parameters for a legged system. The modelconsists of a body mass m

b

at height yb

. Forces delivered tothe mass are modulated by an actuated rack and pinion. Apinion of radius r is driven by an ideal actuator with outputtorque ⌧ . An inertia I represents the total rotational inertia ofthe actuator including gear transmission. The rack, with massm

`

, abstracts the entire leg and interfaces with the groundthrough a Hookean spring with stiffness k

i

. This stiffnessrepresents a lumped stiffness of the ground, foot covering, and

any residual structural stiffness of the leg. We assume that thisinterface spring has no preload. This assumption is capturedin the model by placing the origin in the vertical direction at aheight which corresponds to the spring’s rest length at impact.

Letting q = [yb

, ✓]T , the dynamics of the leg model follow

H ¨q + Kq + g = ST ⌧ (1)

from standard Lagrangian methods, with

H =

hm

b

+m`

�rm`

�rm`

m`

r2 + I

ig =

h(m

b

+m`

)g�m

`

gr

i(2)

K =

hki

�rki

�rki

r2ki

iS = [

0 1] . (3)

A modal analysis of the system is tractable, providing insightinto both the impact dynamics and the mechanical bandwidthof the leg. This simple linear model captures many featurespertinent to the design of multi-DoF legs, such as the effectsof reflected actuator inertia on the GRFs following impact.The ground reaction force f(t) onto the leg is provided bythe interface spring

f(t) = �ki

y`

(t) = �[ki

� rki

]| {z }:=C

q(t) . (4)

Letting s be the Laplace variable, the Laplace transformof the ground reaction force F(s) is related to the Laplacetransform of the torque T (s) and initial conditions through:

F(s) = C(s2H + K)

�1⇥ST T (s) � g/s + H(sq0 +

˙q0)⇤

(5)To analyze properties of the GRF, drop-test conditions areassumed. The leg is assumed stationary with respect tothe body at an impact with initial velocity y

b,0. As such,q0 = [0, 0] , and ˙q0 = [y

b,0, 0]

T. The Laplace transform ofthe ground reaction force F(s) can then be found throughalgebraic expansion of (5):

F(s) =

rki

mb

d(s)T (s)+

ki

(mb

m`

r2 + Imb

+ Im`

)

d(s)

⇣�y

b,0 +

g

s

⌘(6)

where

d(s) = (mb

I + m`

I + r2mb

m`

) s2 + ki

(mb

r2 + I) . (7)

This second-order system has a single natural frequency !n

=pk

i

/me

where me

is the effective mass felt by the spring

me

=

mb

I + m`

I + r2mb

m`

mb

r2 + I. (8)

It is interesting to note that the reflected actuator inertiadirectly modulates the effective mass

lim

I!0m

e

= m`

, and lim

I!1m

e

= m`

+ mb

, (9)

where any increase in I leads to an increase in me

dme

dI=

r2m2b

(mb

r2 + I)

2> 0 . (10)

Further, letting

J :=

@y

`

@yb

@y`

@✓

�= [1, �r] (11)

Leg Mass (kg)10-4 10-2 100 102

Inertia(kg"m2)

10-8

10-6

10-4

10-2

100

102

104Bandwidth (Hz)

101:5

102

102:5

103

103:5

104

Leg Mass (kg)10-4 10-2 100 102

Inertia(kg"m2)

10-8

10-6

10-4

10-2

100

102

104Max. Impact Force (N)

101:5

102

102:5

103

103:5

104

Fig. 3. Influence of leg mass m`

and actuator inertia I on open-loop forcecontrol bandwidth !

n

and maximum impact force assuming an initial velocityof y

b,0 = �1 m/s. Parameters ki

= 106 N/m, r = 0.1 m, and mb

= 32 kgwere used to roughly match the MIT Cheetah on rigid ground with a 3 mmrubber footpad.

as the Jacobian for leg mass, straightforward algebra verifies

me

=

�JH�1JT

��1. (12)

Thus, me

is precisely a task-space (operational-space) inertia[27] measured at the foot prior to the interface spring. With thisconnection, Section III-B generalizes the analysis to higher-DoF articulated mechanisms.

Returning to the system (6) and examining the impact forcein more detail, we assume ⌧ = 0 in order to isolate the passivemechanical properties of the mechanism. The inverse Laplacetransform of (6) then provides

f(t) = �yb,0

pk

i

me

sin(!n

t) + me

g(1 � cos(!n

t)) . (13)

Figure 3 shows the dependency of the maximum impact forceand mechanical bandwidth !

n

on the leg design parameters Iand m

`

. Due to the dependence of impact force and bandwidthon m

e

, a combination of low leg mass and low actuator inertiasimultaneously maximizes open-loop force control bandwidthand minimizes impact force magnitudes. For high leg mass, theactuator inertia has comparatively less effect on these metrics.

This analysis was extended to the case of including seriescompliance into the pinion actuation, as shown in Fig. 2(b).The addition of compliance helps to soften impact, but isknown to have a detrimental effect on closed-loop forcecontrol bandwidth. In terms of design, the requirements ofa closed-loop controller to fight the natural dynamics ofthe system in high-frequency regimes can be minimized bydesigning to maximize the open-loop mechanical bandwidthof the mechanism. For this extended model, we approximateits mechanical bandwidth with its lowest natural frequency.

In this extended case, the system dynamics can again beplaced in the form of (1). A series spring of stiffness k

s

ismodeled between a motor angle ✓

m

and a spring output angle✓. The extended system thus has configuration q = [y

b

, ✓, ✓m

].

Series Spring Constant ks (Nm/rad)101 102 103 104 105

Fre

quen

cy(H

z)

100

101

102

103 !1

!2

!n

~!1

~!2

!2 =q

ks/ Ir2mbm`r2mbm`+I(m`+mb)

!1 =q

ks/ Ir2mbI+r2mb

Fig. 4. Influence of series spring stiffness ks

on the natural frequenciesof the system, !1 and !2. The mechanical bandwidth, as determined by thelowest natural frequency, increases with k

s

. Parameters r = 0.1 m, I =0.010 kg ·m2, k

i

= 106 N/m, m`

= 3 kg, and mb

= 32 kg, roughly matchthat of the MIT Cheetah running on rigid ground with a 3 mm rubber footpad.

The system kinetic T and potential V energies take the form:

T =

1

2

mb

y2b

+

1

2

I ˙✓2m

+

1

2

m`

y2`

, and (14)

V = g mb

yb

+ g m`

yl

+

1

2

ki

y2`

+

1

2

ks

(✓m

� ✓)2 . (15)

A Lagrangian development can be followed to derive thedynamics, again resulting in a linear system (1).

Following this development, the ground reaction force forthe extended system again matches (5). The torque-to-forcetransfer function H

⌧

(s) satisfies

H⌧

(s) = C(s2H + K)

�1ST

=

ks

rki

mb

↵s4 + �s2 + �(16)

where

↵ = Ir2mb

m`

, (17)� = k

s

(Imb

+ Im`

+ r2mb

m`

) + Ir2ki

mb

, and (18)� = k

s

ki

(mb

r2 + I) . (19)

Letting j represent the imaginary variable, the four poles s =

±j !1,2 of (16) provide the two natural frequencies of thissystem as

!1,2 =

s� ⌥

p�2 � 4↵�

2↵. (20)

Graphs of these two frequencies are shown in Fig. 4. Asintuition may suggest, !1 ! !

n

as ks

! 1, where !n

isthe natural frequency for the previous prototype with rigidtransmission. At low transmission stiffnesses k

s

, the interfacespring k

i

and leg mass m`

do not affect the mechanicalbandwidth

!1 :=

qks/ Ir2mb

I+r2mb⇡ !1 (21)

At high stiffnesses ks

, a similar, yet less physically meaningfulresult holds for !2

!2 :=

qks/ Ir2mbm`

r2mbm`+I(m`+mb)⇡ !2 . (22)

Although it is possible to achieve a closed-loop force controlbandwidth beyond the lowest natural frequency of the system,this process is sensitive to model-based information and re-quires additional actuation effort to fight the natural dynamics

of the system. This is further complicated in applications ofnon-collocated force control which must compensate for thedetailed dynamics of potentially many transmission elementsbetween the sensor and the actuator.

Thus, we argue that the open-loop mechanical bandwidthof a mechanism ultimately limits the practical closed-loopbandwidth. Indeed, with the numbers used in the figure above!

n

= 79.7 Hz, a value similar to that reported with our frame-work in Section V. The addition of a stiffness k

s

= 70 Nm/rad,similar to that in StarlETH [26], provides !1 = 13.5 Hz,similar to the 9 Hz closed-loop bandwidth reported in theirLQR-based SEA torque control.

Through this analysis, it follows that to minimize impactforces and maximize mechanical bandwidth, a design shouldbe sought with minimal reflected actuator inertia, minimal legmass, and minimal actuator compliance. Reflected inertia inparticular has been shown to play an important role in theeffective mass that governs collisional dynamics. The nextsection will address how to generalize and quantify theseeffects in more complete leg models.

B. The Impact Mitigation Factor (IMF)

In more complex mechanisms, factors such as actuatorplacement and the structure of the leg articulation ultimatelydetermine how reflected actuator inertias affect the back-drivability of the robot. While backdrivability includes bothvelocity-dependent and inertia-dependent effects [2], inertia-dependent effects are much more difficult to shape throughclosed-loop impedance control [16], [28]. Thus, these passiveinertial effects are purely dependent on the inherent designcharacteristics of the robot. Of these inherent characteristics,reflected actuator inertias play an important role. They directlydetermine the degree to which the body inertia contributes toimpacts, as observed in the simple leg model in (9). As aresult, the highest loads felt in the legs, gearboxes, and othertransmission components are governed, in large part, by designdecisions centered on reflected actuator inertia.

Previous work has addressed the role of the effective contactinertia in modeling impact events [29], which is used inSection III-B1. This previous work modeled the values ofimpact impulses under specific impact conditions, which maybe of use in comparing designs for a specific robot. For un-structured environments where impact conditions may not beknown, however, maximizing overall inertial backdrivability isimperative to mitigate impact forces.

With these impact-related motivations, this section quanti-fies how effectively the free dynamics of the mechanism areat reducing impact impulses in a floating-body robot. To em-phasize the importance of actuator design on impact dynamics,impulse reduction is evaluated through comparison to a designwith worst-case reflected inertia, one where all joints arerigidly locked. Through this approach, a new metric, called theimpact mitigation factor (IMF), is introduced to quantify thenormalized inertial backdrivability of the mechanism. Despitethe importance of backdrivability to physical interaction, thereare not yet meaningful metrics to compare backdrivabilityacross machines and to include EM actuator properties in

such a comparison. The new development in Section III-B2illuminates the effects of actuator design on previous impactanalyses, while Section III-B3 formulates the new IMF metricto quantify inertial backdrivability across different robots.

1) Rigid-Body Impact Dynamics: Given a floating-bodysystem, with base coordinates q

b

2 R6 and internal (joint)coordinates q

j

2 Rn its dynamics can be compactly describedthrough

H

bb

Hbj

Hjb

Hjj

� ¨q

b

¨qj

�+ h(q, ˙q) = ST ⌧ + JT f (23)

where q = [qb

,qj

]

T 2 Rn+6, h(q, ˙q) 2 Rn+6 includes theCoriolis, gravity, and spring-dependent terms, J 2 Rm⇥(n+6)

is a contact Jacobian, and f 2 Rm represents the contact force.For point-foot contacts considered here, m = 3. In a case ofseries compliance at the joints, q

j

may include both joint andmotor angles. In contrast to the previous section, we do notmodel any ground compliance, and assume that impact can beconsidered as an impulsive event. As a result, the analysis isidealized in comparison to that in the previous section.

Given a state q, ˙q just before a foot impact, the system hitsthe ground with a velocity v = J ˙q and experiences a contactimpulse ⇢ 2 R3 as given by [29]:

⇢ = �⇤v . (24)

⇤ 2 R3⇥3 is the operational-space inertia matrix (OSIM) [27],[30], [31] of the system felt at the contact and is given by

⇤ =

�JH�1 JT

��1. (25)

2) Actuator Effects on the OSIM: It is important to notehow the reflected inertias of EM actuators affect the OSIM.Generally, the mass matrix H, as partitioned in (23), providesa kinetic energy metric

T =

1

2

˙qTH ˙q (26)

where T includes the kinetic energy of rigid-body links as wellas rotor inertias and gears of EM actuators. Thus, the massmatrix can always be partitioned into a matrix that accountsfor the kinetic energy of all rigid-body links H

rb

and a matrixH

mot

that accounts for the kinetic energy of the moving partswithin link-mounted actuators (rotors, gears, etc.)

H = Hrb

+ Hmot

. (27)

Hmot

is often approximated as [32]

Hmot

= diag(01⇥6, I1, . . . , In

) (28)

where Ii

represents the reflected rotational inertia on DoFi and scales as the square of its associated gear ratio. Thisapproximation is only valid for actuators with large gear ratios,where the only significant kinetic energy of the motor elementsis from rotational kinetic energy along their rotational axes.Regardless of this approximation, it can be seen from

⇤ =

⇣J (H

rb

+ Hmot

)

�1JT

⌘�1(29)

that any increase to the actuator inertia Hmot

(in a positivesemidefinite sense) necessarily gives rise to an increase incontact inertia ⇤, generalizing the simplified result in (10).

In the worst-case scenario, if reflected inertias becomearbitrarily large on each joint, the system will roughly behaveas if all the joints are locked. This worst-case design canbe used as a benchmark to evaluate the effects of reflectedinertias in designs with the joints free to move. Partitioningthe Jacobian as

J =

⇥Jb

Jj

⇤, (30)

an impact velocity v would give rise to a contact impulse ina locked system ⇢

L

as

⇢L

= ��Jb

H�1bb

JT

b

��1v . (31)

With this insight, we define a locked contact inertia ⇤L

as

⇤L

=

�Jb

H�1bb

JT

b

��1. (32)

Intuitively, the apparent inertia of the system with joints lockedshould be greater that the inertia felt with the joints free torotate. This can be shown more formally, that ⇤

L

� ⇤ in apositive semidefinite sense. Proof is given in the appendix. Itis important to note that ⇤

L

is invariant with changes in gearratio (assuming negligible change in overall mass).

3) Impact Mitigation: Development and Metric: Given animpulse that is experienced in the locked system, it is naturalto consider: How effectively are the free dynamics of themechanism at mitigating this impulse? That is, assuming thesame impact velocity, how would the impact in the free systemcompare to its non-backdrivable equivalent? Given the lockedimpulse ⇢

L

, equal impact velocities v are achieved when thefree system experiences an impulse

⇢ = ⇤⇤�1L

⇢L

(33)

Roughly, the term ⇤⇤�1L

characterizes how much inertia isfelt at impact in comparison to the overall (locked) systeminertia. The reduction in impulse is then given by

⇢L

� ⇢ =

�I�⇤⇤�1

L

�| {z }

:=⌅

⇢L

. (34)

where ⌅ is introduced as an Impact Mitigation Matrix (IMM),and its determinant

⇠ := det(⌅) (35)

as the Impact Mitigation Factor (IMF). This new factor hasinteresting properties that pertain to backdrivability. It is shownin the appendix that 0 ⇠ 1. An IMF ⇠ = 1 corresponds toa system with perfect inertial backdrivability that eliminates allimpact, whereas ⇠ ! 0 as ⇤ ! ⇤

L

. Due to the normalizationprovided from the locked case (with roughly infinite reflectedactuator inertia), the IMF is a nondimensional quantity whichenables comparison across machines of different scale.

Further, one may be interested in the capability of a designto mitigate impact in a particular direction, for instance inthe case of a hopping robot subject to predominantly vertical

impacts. Given a direction x, we define the directional IMF(DIMF) ⇠

x

as

⇠x

= 1 �12x

T⇤x12x

T⇤L

x(36)

where ⇠x

similarly inherits the property that 0 ⇠x

1

from the fact that ⇤L

� ⇤. This DIMF has a clear physicalinterpretation as well. The kinetic energy lost due to an impactwith velocity v is

�T =

1

2

v⇤v . (37)

Thus, ⇠x

quantifies the percent decrease in energetic lossesafforded through the free dynamics of the mechanism whenimpacting in direction x.

4) Series Elasticity and the IMF: When compliance isadded to a stiff transmission model, portions of the actuatorinertia are no longer inertially coupled to the foot. This inertialdecoupling affects ⇤ and the associated impact impulse. Thecompliance, roughly, redistributes some of the impact impulseto a window of time following impact. Changes in the valuesof the stiffness modify how spread out the forces become, butdo not change the initial impulse, which is governed by inertialproperties alone. In this sense, adding compliance to a rigidtransmission model will affect the IMF, while changes to thevalues of stiffness will not. Considering non-impulsive impactevents over a finite window of time, which should always bethe case for collisions in reality, represents an important areafor future extension of the IMF. Further details on the IMF fora system with SEAs are provided in comparison to the designof the MIT Cheetah leg in Section V.

C. Geometric considerations for electromagnetic motors

From the analysis of the previous section, we seek to deter-mine principles for actuator selection that minimize actuatorinertia while providing the high torques necessary for high-speed locomotion. Many design parameters of EM actuatorscontribute to overall performance. However, we consider gapradius to be a critical parameter for the purposes of maximiz-ing torque density and providing transmission transparency.The gap radius of an EM actuator is the radius of the magneticinterface between the rotor and the stator. While gap radiusis one of many possible condensed indicators of performance[33], it is directly related to torque density and torque per totalinertia. Torque per total inertia, in turn, is directly related tothe available bandwidth of the actuator.

Ignoring edge effects, the axial length of the motor does notaffect torque density and torque per inertia because increasingaxial length is equivalent to adding identical motors on thesame axle. In Figure 5, Emoteq HT series motor characteristicsare plotted against gap radius. Motors of the same gap radiusand various lengths have overlapping values of torque densityand torque per inertia, whereas gap radius directly affects thosecharacteristics. For extreme geometries beyond these designs,edge effects degrade the metrics as length decreases.

1) Effects of Geometry on Motor Performance Metrics:To understand the main effects of motor geometry on torquedensity and torque per inertia, we consider a class of designs

100.5 101 101.5 102

Gap Radius rg (mm)

10-6

10-4

10-2

100

102

104

106

==J / rg!1:6

==M / rg0:8

K2t =R / rg

4:1

Fig. 5. Motor performance metrics ⌧/J , ⌧/M , and K2t

/R extracted fromEmoteq frameless HT series motors.

under the following simplifying conditions. First, the radialthickness of the rotor and stator are assumed fixed. Second, thecylindrical geometries of the rotor and stator are approximatedto first order as thin walls. As a result, the actuator massis given by M = 2⇡r

g

l(ts

⇢s

+ tr

⇢r

) and rotor inertia byJ = 2⇡lt

r

⇢r

r3g

, where rg

is the gap radius, l is the axiallength, and t⇤ and ⇢⇤ are the radial thickness and density,respectively, of the stator and rotor. Third, current density isassumed constant [34]. Neglecting edge effects, this resultsin conditions of thermal and electromagnetic similarity, wherethe steady-state stator temperature and average shear stress �

s

on the rotor are constant across designs. The resulting torquefrom this stress ⌧ = 2⇡r2

g

l�s

does depend on motor geometry.Therefore, within this class of designs, we predict that

torque density and acceleration capability (torque per rotorinertia) scale by the following relationships with r

g

, and weobserve no effect from axial length.

⌧/M / rg

(38)

⌧/J / 1/rg

(39)

Another important characteristic is torque production effi-ciency, which is related to the motor constant K

M

. K2M

isequivalent to the torque squared per unit ohmic power loss.Torque is generated from sheer stress �

s

/ IBn/A where Iis the motor current, B is the field strength of the magnets, nis the number of wires in a cross section perpendicular to theaxis, and A / r

g

is the area of the cross section. Using thisrelationship, torque production efficiency is given as

K2M

=

K2t

R=

⌧2

I2R/

r2g

l2B2n2Aw

⇢lw

(40)

where Kt

is the torque constant, R is the terminal resistance,A

w

is the cross section area of each wire, lw

the total lengthof wire, and ⇢ is the the resistivity of the wire material.

An additional set of assumptions is used to simplify thisrelationship. First, a wire length l

w

/ nl is used, whichassumes all stator wire contributes to winding coils. Second,a fixed wire gauge is assumed, which implies that the number

tr

ts

rg

lStator

Rotor

range of motor geometries

rg

: gap radius

l : motor length

tr

: rotor thickness

ts

: stator thickness

M : mass

Assumptions:

tr

, ts

, M = constant

Fig. 6. Motor performance tradeoffs are considered for a constant mass bud-get under assumptions of constant rotor/stator thickness and electromagneticsimilarity.

of wires in the cross section n / rg

. Thus, the relationshipbecomes

⌧2

I2R/ r3

g

l . (41)

2) Comparison to Catalog Data: Figure 5 shows valuesfor these metrics across data collected from Emoteq HTseries motors. The max continuous torque as reported on datasheets was used to approximate conditions of electromagneticsimilarity. Within this dataset, the torque density was foundproportional to r0.8

g

; torque per rotor inertia proportional tor�1.6g

; and torque production efficiency proportional to r4.1g

.These factors of proportionality, r0.8

g

, r�1.6g

and r4.1g

arerelatively consistent with the proportions in our dimensionalanalysis r1

g

, r�1g

, and r3g

from (38), (39), and (41) respectively.The similarity is to be taken considering that the dimensions

of these motors used in Fig. 5 do not exactly match ourassumptions, and in particular the assumption of constantstator and rotor thickness. In the Emoteq HT series motors, thestator and rotor thicknesses scale by t

s

/ r0.8g

and tr

/ r0.4g

respectively. The incorporation of these trends into the pre-vious analysis would lead to a lower ⌧/J , lower ⌧/M , andhigher K2

t

/R as observed in the motor catalog data. Changingrotor/stator thickness, however, has a nonlinear effect on themagnitude of the magnetic field at the rotor-stator air gap,and heavily influences stator winding design. As a result, theeffects of these changes are difficult to model accurately ingeneral. The remainder of the section returns to considerationsof fixed rotor and stator thickness across designs.

3) Optimizing Geometry Under Design Constraints: Thetheoretical motor performance metrics (38) and (39) can behelpful in making initial decisions related to actuator geometrywhile considering design constraints. Such considerations areunique relative to previous isometric scaling analyses thatfixed geometry [34], [35]. Suppose a desired output torque⌧out

and a fixed mass budget M for an actuator in the classconsidered previously. Under this mass budget, motor lengthis constrained to scale by l / 1/r

g

as shown in Fig. 6. Tomeet the output torque requirements across different designs,suppose that a gear train with ratio N is used.

As an initial approximation, assume that the gear box hasnegligible added mass, inertia, and friction torques from thegears themselves. The required gear ratio as a function of gapradius is then

N =

⌧out

2⇡r2g

l�s

/ 1/rg

(42)

with a reflected rotor inertia through the gearbox of

Jref

= Jr

N2= 2⇡r3

g

ltr

⇢r

N2 / r0g

(43)

when the scalings for l and N are considered. While the totalreflected rotor inertia and the total torque at the output shaftstay the same, increasing the gap radius lowers the requiredgear ratio. In summary, in a design space where the motormass and the output torque requirement are held constant, andgearboxes are idealized, (43) shows that total reflected inertiaremains constant across variations in gap radius.

Considering the non-idealized effects of gear transmissions,however, shows the benefits of high-gap radius designs whenthe actuator mass budget is fixed. The results of constantoutput torque and constant reflected inertia only hold acrossdifferent design geometries if the mass, friction, and inertiaof the gears are ignored. If we assume instead that mass,friction, and gear inertia increase monotonically with gear ratioN , this implies in turn that ⌧

out

will decrease and Jr

willincrease monotonically with N . Thus, these considerationsfavor a larger gap radius motor when mass budget is fixed.Such a design will have a smaller gear ratio, fewer gear stages,and less gear mass, resulting in less friction loss, higher torquedensity, higher bandwidth, and higher IMF.

Therefore, in this design space, there is no tradeoff, and inits purest form, this analysis advocates for high-gap-radiusdirect-drive robots [36]. However, the geometry required fordirect drive to have enough torque is typically infeasible. Forexample, in order to design a direct-drive motor for the MITCheetah, one such motor would have been 76.2 cm in diameterwith a 5 mm axial length. Considering these limitations, theoptimal actuator for a given mass will thus consist of a motorwith the largest gap radius as allowed by space and the smallestgear ratio as required by torque specifications.

4) Other Considerations: For extreme geometries, edgeeffects begin to degrade the benefits of high-gap-radius de-signs. As a result, it is important to view this analysis asproviding a guiding principle to focus design pressure foractuators in legged machines. Outside the class of motorsconsidered, or at edges of the design space, detailed EM,winding, and manufacturing considerations should limit thedegree to which gap-radius effects are regarded as dominantwithout further modeling.

In addition to motor geometry optimization, we can increasetorque density by utilizing the intermittent torque of themotor, which is much higher than the continuous torque limit.These operating regimes are depicted in Fig. 7. Many roboticactuators operate under the continuous torque limit, althoughthe duty factor of high torque usage in most legged robots issmall. In legged locomotion, the continuous torque limit of themotor does not limit torque capability. Unlike most EM motorapplications, the joint torque profiles in legged locomotionconstantly fluctuate.

IV. PROPRIOCEPTIVE ACTUATION IN THE MECHANICALDESIGN OF THE MIT CHEETAH LEG

The MIT Cheetah is designed to emulate various locomotioncapabilities of quadrupeds such as walking, running, and jump-ing. Such behaviors involve repeated high impacts followed by

τ"

0 ω"

Demag. torque

operation range recommended by motor manufacturer

Cont. torque

~0.3kW/kg

Sat. Torque

Continuous operation region

Intermittent Operation region

Demagne'za'on* Torque

Satura'on Torque

Con'nuous Torque

Fig. 7. Different operational limits of a motor in torque-speed space.The green area represents the manufacturer-recommended operation that isbounded by the voltage limit and continuous torque determined by the thermallimit. The blue area shows the extended operating range of the motor byraising voltage and speed but maintaining current under the continuous torquelimit. The motor may intermittently exceed the continuous torque limit andoperate in the transient operation space shown in peach color. Beyond thesaturation torque, parts of iron in the stator become magnetically saturatedand the torque/current relationship becomes highly nonlinear as the torqueconstant decreases. This nonlinearity can be compensated for in control toprovide accurate torque delivery. At the demagnetization torque, magneticfields from the stator begin to demagnetize the rotor magnets, damaging themotor.

Knee Stator

Knee Rotor

Base Planetary gear train (1:5.8)

Output to linkage

Encoder mount

Output to hip

Hip Rotor

Encoder mount

CoM Hip Stator

Fig. 8. Shoulder module section view for a front leg of the MIT Cheetah.

short ground contact times. For example, the ground contacttime of each leg during 6 m/s bounding is around 60 ms.In order to control the ground reaction forces during such ashort period of time, the system must have high-bandwidthforce control and survive frequent high impact forces. TheMIT Cheetah is the first embodiment of the proprioceptiveactuation concept in legged locomotion.

Figure 8 shows the detailed design of the leg module.The four hip modules of the robot are identical with minordifferences in segment length ratios among the three links andbetween the front and rear legs. As discussed in Section III-C,it is critical to maximize the ratio of torque to inertia. Weselected design parameters for motor with the largest radiusthat could fit within a 12.5 cm diameter space, which isconstrained by the size of the MIT Cheetah. The first versionof the MIT Cheetah, which trotted up to 6 m/s [13], usedan Emoteq HT-5001 frameless motor. The second version

of the MIT Cheetah, capable of bounding outdoors, usescustom motors designed by the MIT team [37]. The motorwas specifically designed to maximize the saturation torquefor a given mass, where the major design tradeoff is betweenohmic loss and the saturation torque. The saturation torquedensity of the custom designed motor is around 27 Nm/kg(standalone without module or gearing), significantly higherthan 9 Nm/kg in the Emoteq HT-5001. Given this high torquedensity, a 3 cm motor length is able to be used on each axis.The rotor inertia of the custom motor is three times that of theEmoteq HT-5001 and the attached leg inertia is slightly largerthan the previous MIT cheetah.

Given the large torque capacity of the motor, we chosethe gear ratio (1:5.8)1 to meet the normal GRF generationrequirements for running at a range of speeds up to 13.5 m/s.We chose a one-stage planetary gear train with four planetgears. Such a low gear ratio provides a higher IMF, beneficialfor highly-dynamic locomotion. Unlike traditional serial-linkrobots in which actuators are present at every joint, twoactuators and the gear train are coaxially located at the hipof the leg to minimize the total moment of inertia, whichallows for a compact and robust design. One of these actuatorsdirectly actuates the hip, while the other actuates the kneethrough a parallel linkage. The structure of the leg is alsodesigned to minimize mass and leg inertia, and thus maximizethe impact mitigation factor. Bending stress is minimized in theleg structure by distributing tensile forces to the tendons. Thismethod allows significantly lower inertia of the leg withoutcompromising leg strength [38]. The shoulder module containsthe motors (2.00 kg), heat-treated steel gear trains (0.42 kg),and framing (1.23 kg) for a total mass of 3.65 kg. A leg withthree ABS plastic links is attached to each module, with acombined mass of 850 g. The lightweight leg allows the centerof mass to be located inside of the shoulder module. As aresult, a small moment of inertia of the leg allows for rapidmovement and high-bandwidth force control.

V. ASSESSMENT AND RESULTS

A. Impact Mitigation Factor

In order to assess the backdrivability of the MIT Cheetahleg design, the impact mitigation factor (IMF) was computedand compared to other legged systems. As noted in SectionIII-B, the IMF is configuration dependent. This configurationdependency is shown visually in Fig. 9. The figure shows thedirectional IMF (DIMF) ⇠

x

for directions x in the sagittalplane. For the Cheetah, the IMF correlates most strongly withknee angle, as shown in Fig. 10. Due to the large mass andinertia of the components within the Cheetah’s motors, (28)was not used for simplification in the calculation of the IMF.The motor rotors, for instance, have rotational inertias on out-of-plane principal axes that are the same order of magnitudeas for the leg links. These inertial contributions would beotherwise ignored through the use of (28). The IMF numbersaccount for the mass and inertias of the motor rotors and

1This gear ratio was also driven by practical considerations, as it is thelargest single-stage ratio available for a standard planetary configuration withcommodity English geartooth options.

245 mm

220 mm

182 mm

18 mm

82 mm

84 mm m3 = 0.116 kg

I3 = 7.4 kg cm2

m2 = 0.157 kg

I2 = 13.1 kg cm2

m1 = 1.55 kg

I1 = 80.9 kg cm2

Femur

Tibia

Foot

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

zy

Fig. 9. Visual description of the configuration dependence of the impactmitigation factor (IMF), and main dynamics parameters for a leg of the MITCheetah. The dashed line represents an ideal directional IMF (DIMF) of 1,while the blue represents the computed DIMF.

Knee Angle (rad)0 1 2 3

IMF9

0

0.2

0.4

0.6

0.8

1

Knee Angle (rad)0 1 2 3

9 x

0

0.2

0.4

0.6

0.8

1

Knee Angle (rad)0 1 2 3

9 y

0

0.2

0.4

0.6

0.8

1

CheetahSEA CheetahStarlETHHuboSEA Hubo

Knee Angle (rad)0 1 2 3

9 z

0

0.2

0.4

0.6

0.8

1

Fig. 10. Impact mitigation factor and directional impact mitigation factorsversus knee angle. The coordinate system for these numbers is shown in Fig. 9for the Cheetah with similar coordinate systems relative to the sagittal planefor the other morphologies.

planetary gears, as estimated from CAD models. The rotorhas inertia 3.0279 ⇥ 10

�4 kg m2 along its rotational axis,with a 5.8:1 gear reduction, resulting in a reflected inertia of0.0102 kg m2. Other main parameters of the leg are given inFig. 9 for reference.

The IMF of the Cheetah design was also compared withapproximate StarlETH and HUBO Plus models. Rigid-bodymasses, CoM locations, and kinematics for StarlETH weretaken from an available publication [26]. Previous work pro-vided estimated rotational inertias in the sagittal plane [39]. In-ertias in other directions were computed from an equi-densityassumption on a bounding box estimated from graphical mod-els. No reflected actuator inertia was modeled for StarlETH, asits series elastic actuators decouple reflected inertia from theendpoint. The HUBO model was taken from a URDF of the

HUBO Plus robot [40]. The DC actuators assumed at the jointshad a rotor inertia of 3.33 ⇥ 10

�6 kg m2 with a 160:1 gearreduction, producing a total reflected inertia of 0.0852 kg m2.No reflected inertia effects of the gear reduction were modeled.IMFs were computed assuming impact in the center of thebottom of the foot.

Figure 10 shows that the SEA-actuated StarlETH has thehighest impact mitigation factor over a range of knee an-gles. For these computations, the virtual leg in each systemremained upright, while the knee angle was used to modulatevirtual leg length. Low IMFs for small knee angles are inmost part due to reduced backdrivability in the ˆz direction, asindicated by ⇠

z

. A similar degradation in ⇠x

is observed inStarlETH and Cheetah when the leg becomes fully collapsedat a knee angle of ⇡ rad. The IMF of HUBO experiencesless peak-to-peak variability in IMF across knee angle. Thisdifference can be attributed to the additional articulation inthe HUBO hip and ankle which further prevent its upper-bodymass from being felt at impact even for small knee angles.

To assess the relative contribution of reflected actuatorinertia to the IMF of HUBO and Cheetah, hypothetical modifi-cations of these machines were considered that included SEAsat the joints. Averaged over the configurations considered, theIMF of unmodified Cheetah was 90% that of the hypotheticalSEA Cheetah. In comparison, the IMF of unmodified HUBOwas 52% that of a hypothetical SEA HUBO. HUBO’s designdoes include significant actuator mass distally in the leg, whichin part causes this difference.

B. Step-input test on the MIT Cheetah leg

To initially evaluate proprioceptive force control, the Chee-tah robot leg was mounted inside of an axial material testingdevice (Zwick Roell BX1-EZ005.A4K-000). Its stock forcesensor was replaced with a six-axis force-torque sensor (ATIdelta, SI-660-60 calibration) which can measure up to 1980 Nin the z-axis with 0.25 N resolution.

To identify the open-loop force control bandwidth of theleg, a 100 N step-input test was executed. A pure radialforce was commanded and mapped to desired leg torquesassuming static loading conditions. Desired joint torques werethen realized using closed-loop motor current control withcustom motor drivers and current control taking place at arate of 20 KHz. The currents from these drivers were used toestimate the force at the foot, again under an assumption ofstatic loading conditions. Fig. 11 shows the results of this test.In this figure, the black line is the commanded force, the redline is the estimated force from measured motor currents, andthe blue line is the force measured by the external sensor. Thediscrepancy between the estimated force and the exteroceptivesensed force highlights the presence of structural compliancein the leg. This structural compliance effectively decreases thebandwidth of the proprioceptive force control below that ofthe electrical dynamics. Assuming a second-order response,the response bandwidth of the leg system can be estimatedbased on the measured rising time, as seen in Table I.

Further details on similar experiments with the MIT Cheetahleg can be found in our previous publication [5]. This previous

Step Input Test (100N, Radial Direction)

Page 11

160

0

20

40

60

80

100

120

140

Time(ms) 100 0 10 20 30 40 50 60 70 80 90

Forc

e(N

)

From 𝑓 × 𝑡 = 0.35

Driver & Motor Current Bandwidth: 400Hz Leg System Bandwidth: 103.7Hz Stabilization Time: 55 ms

To fully control 400Hz Bandwidth System, control frequency should be at least 800Hz. Usually use x5 to x10 for good control.

Target: 4kHz control frequency

(250us loop time)

Estimated from Motor

Force Sensor Commanded Force

Force Sensor Commanded Force

Estimated from Motor

Time (ms)

Force(N)

Fig. 11. Step-input test on the Cheetah robot leg. The force estimated fromthe motor is determined from the measured current in the stator windingsusing an assumption of static force transfer from the actuator to the foot.

System Rise time Tr

(ms) Bandwidth (Hz)motor driver to winding current 0.875 400

motor driver to foot force 3.38 103.7

TABLE ISYSTEM BANDWIDTH ESTIMATION FROM STEP-INPUT TEST.

work demonstrated the capabilities of proprioceptive forcecontrol to realize a virtual stiffness at the foot. The nextsubsection goes beyond impedance control and demonstratesopen-loop force control for more general force profiles.

C. Proprioceptive Force Control Test

To investigate the performance of proprioceptive forcecontrol for dynamic locomotion, we performed in situ forcecontrol tests during unconstrained 3D bounding. The robotbounded in free space using a control law from previouswork [41]. The bounding controller would output desiredforces f

pro

2 R3 for each stance leg, to be realized throughproprioceptive actuation. These desired forces were mappedto joint torques through

⌧d

= S`

JT fpro

. (44)

Here J 2 R3⇥18 is a foot Jacobian and S`

2 R3⇥18 is aselector matrix for the joints in the leg. The desired torque⌧

d

2 R3 contains torques for the two proprioceptive actuatorsnominally in the sagittal ˆx-ˆz plane, as well as for a moretraditional hip ab/ad actuator which nominally affects thelateral ˆy forces. Coordinate systems follow those in Fig. 10.The ATI delta force-torque sensor was embedded into thecontact surface such that the front left leg of the robot couldrun on the sensor without interference to the gait. Figure 12shows a figure of this experimental setup.

Figures 13-15 show the proprioceptive force control track-ing in the sagittal plane where the proprioceptive actuatorshave force control authority. The force control figures showhigh duty ratio bounding (Fig. 13) in comparison to mediumand low duty ratio bounding (Figs. 14 and 15). Lower dutyratios require shorter contact times down to 85 ms in the mostextreme case considered here. Again, we emphasize that asidefrom statics (Eq. 44), no model-based information was used to

Fig. 12. Experimental setup for proprioceptive force control measurements.The ATI delta force/torque sensor is embedded into the contact surface sothat the front left leg could step on the sensor without interference to the gait.

Time (ms)0 100 200 300 400 500 600 700 800 900 1000

For

cef x

(N)

-150

-100

-50

0

50

Time (ms)0 100 200 300 400 500 600 700 800 900 1000

Forc

ef z

(N)

0

100

200

300

400

500 ProprioceptiveMeasuredEvaluation Period

Fig. 13. Force tracking for duty ratio D = 0.579 bounding.

translate desired force into commanded motor current acrossany of these results.

Table II quantifies the average tracking accuracy of theproprioceptive force control approach. Force control accuracywas evaluated following each impact transient, during theperiods shown in gray in Figures 13-15. These periods startwhen the measured vertical force crosses the proprioceptiveforce, and end when the proprioceptive force goes to zero. Ingeneral, higher duty ratio bounding results in lower averageerrors.

High duty ratio bounding in Fig. 13 shows one of thereasons to pursue proprioceptive force control. The verticalforce of the first hop in particular, shows a low-frequency(⇡20 Hz) unforced oscillation in the vertical direction. Thisadditional vibration mode is caused by unmodeled dynamics(compliance) of the leg structure and mechanisms. Futuredesign efforts may be directed towards further increasingthe stiffness-per-mass ratio of the legs in order to raisethe natural frequency of the structural dynamics. It is theseunmodeled transmission dynamics that limit the bandwidth ofnon-collocated force control schemes.

VI. DISCUSSION

The proprioceptive actuation paradigm offers a number ofqualitative benefits which extend beyond the analysis pre-sented. While designing for high IMF has been motivated

Time (ms)0 100 200 300 400 500 600

Forc

ef x

(N)

-150

-100

-50

0

50

Time (ms)0 100 200 300 400 500 600

Forc

ef z

(N)

0

100

200

300

400

500ProprioceptiveMeasuredEvaluation Period

Fig. 14. Force tracking for duty ratio D = 0.410 bounding.

Time (ms)0 100 200 300 400 500 600

For

cef x

(N)

-150

-100

-50

0

50

Time (ms)0 100 200 300 400 500 600

For

cef z

(N)

0

100

200

300

400

500

ProprioceptiveMeasuredEvaluation Period

Fig. 15. Force tracking for duty ratio D = 0.298 bounding.

to reduce impact forces, high IMF designs indicate favorableactuator characteristics more broadly related to dynamic in-teractions with the environment. High IMF designs capturean ability to transparently control the interaction forces, andto protect both the environment and the robot in instances ofcollision. These benefits have practical importance, improvingthe lifetime of fragile transmission components, reducing theneed for bulky structural designs, and reducing leg dynamicswhich complicate swing-leg control.

As shown in the previous section, the IMF is also tightlyrelated to improved bandwidth in proprioceptive force control(closed-loop torque control with open-loop ground reactionforce control). In dynamic locomotion, foot force controlin short stance times is critical for balance control. Non-collocated force control lacks robustness to unmodelled high-

Duty Ratio (D) Avg. Err. fx

(%) Avg. Err. fz

(%)0.579 24.35 18.200.410 27.54 14.190.298 33.52 22.91

TABLE IIPROPRIOCEPTIVE FORCE TRACKING ACCURACY VERSUS DUTY RATIO

(D). GENERALLY, TRACKING IMPROVES WITH HIGHER D.

Fig. 16. The ability to mitigate impacts and manage high-bandwidth contactinteractions has been an important capability in results beyond those presentedhere, such as the execution of a running jump over large obstacles [43].

frequency dynamics incited around impact, causing contactinstability [3]. The ability to stably make and break contactthough proprioceptive force control motivates broader appli-cation of this paradigm for high-force interaction.

Increasing the torque density of a EM actuator itself is criti-cal to achieve conflicting specifications for high IMF and highefficiency. Since the gear ratio is a critical design parameterin the tradeoff between the torque generation efficiency andthe IMF, increasing torque density of the motor itself enablesa designer to the reduce actuator gear ratios without compro-mising either of the aspects. Through exploiting this strategy,the MIT Cheetah has already achieved a high efficiency (totalcost of transport of 0.5) rivaling animals in a similar scale[13], [42]. Further research to understand and quantify theexact energetic tradeoffs due to gear transmission selectionsmay yet enable operation beyond the efficiency envelope oflegged animals in nature.

While space and scope have prevented their description,these advances in actuator design for impact mitigation andhigh-bandwidth physical interaction have been paramount to-wards this unique platform achieving recent feats such asrunning at up to 6 m/s and successfully jumping over obstaclesup to 80% of the Cheetah’s nominal leg length [43]. Figure16 shows snapshots from an example jumping motion.

VII. CONCLUSIONS

This paper proposed a new actuator paradigm for high-speedrunning robots, provided analyses to demonstrate the centraltenants of this paradigm, and presented experimental resultsof force tracking tests. While proprioceptive actuation waseffectively implemented in early haptic devices, such as thePHANTOM, a prototype leg model illuminates its benefitsto manage impact and open-loop force control for leggedmachines. Geometric scaling analysis indicates that increasingthe gap radius benefits torque density, which plays a criticalrole in system energetics for locomotion. In order to quantifythe backdrivability afforded through this paradigm, we haveintroduced the Impact Mitigation Factor (IMF) which is appli-cable to robots driven by EM actuators with or without serieselasticity at the joints. A successful implementation of theactuator design principles was shown to allow for high-forceproprioception to deliver desired forces through contact withonly motor current sensing. The experimental results show

the promising performance of the proposed actuator designparadigm by evaluating the force production capabilities of aleg from the MIT Cheetah.

These results encourage broader adoption of proprioceptiveactuators to manage physical interaction for emerging appli-cations in robotics. From disaster response to assistance inthe home, proprioceptive actuators may endow next generationrobots with the ability to stably make and break contact, whileproviding the high-bandwidth and force-critical capabilities toreact in unstructured environments. Through further refinementdirected toward these applications, we believe the proposedapproach will enable the development of actuators that yetexceed the capabilities of biological muscles in every aspect.

ACKNOWLEDGMENT

This research was sponsored by the DARPA M3 programthrough contract number W31P4Q-13-1-0014.

APPENDIXIMPACT MITIGATION FACTOR

In the main body of the text, the impact mitigation factorfor a leg was introduced as

⇠ = det

�I�⇤⇤�1

L

�(45)

where ⇤ 2 Rm⇥m is the contact-space inertia matrix of theleg, and ⇤

L

2 Rm⇥m is the contact-space inertia matrix if allof the joints were locked. In this appendix, we will show that⇤ ⇤

L

and 0 ⇠ 1.Using notation from the main body of the text:

⇤�1 =⇥Jb

Jj

⇤ Hbb

Hbj

Hjb

Hjj

��1"JT

b

JT

j

#. (46)

Then, defining A := (Hjj

�Hjb

H�1bb

Hbj

)

�1 it follows fromthe matrix inversion lemma that

⇤�1= J

⇣H

bb

�Hbj

H�1jj

Hjb

⌘�1�H�1

bb

Hbj

A

�AHjb

H�1bb

A

�JT (47)

= J

H�1

bb

+H�1bb

Hbj

AHjb

H�1bb

�H�1bb

Hbj

A

�AHjb

H�1bb

A

�JT . (48)

Factorizing this matrix,

⇤�1

= ⇤�1L

+ Jh�H�1

bb

Hbj

I

iA

h�H�1

bb

Hbj

I

iT

JT , (49)

where the second term on the right hand side is positivesemidefinite since A is a Schur complement of H. Thus,

⇤�1 � ⇤�1L

and ⇤ ⇤L

. (50)

From this fact, it follows that 0 ⇤� 1

2L

⇤⇤� 1

2L

I. Thus

0 det

⇣I�⇤

� 12

L

⇤⇤� 1

2L

⌘ 1, and (51)

0 det

�I�⇤⇤�1

L

� 1 . (52)

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Patrick M. Wensing (S’09, M’15) received theB.S., M.S., and Ph.D. degrees in Electrical andComputer Engineering from The Ohio State Uni-versity, Columbus, OH, USA, in 2009, 2013, and2014 respectively. He is currently a PostdoctoralAssociate in the Biomimetic Robotics Laboratorywithin the Department of Mechanical Engineeringat the Massachusetts Institute of Technology.

His research interests lie at the intersection ofdynamics, optimization, and control as applied toagile, intelligent, and physically interactive robotic

systems. He was awarded an NSF Graduate Research Fellowship for hisdissertation research on balance control strategies for humanoid robots. AtMIT, the results of his postdoctoral work on the MIT Cheetah 2 robot havereceived considerable publicity worldwide, with features from TIME, WIRED,and the Wall Street Journal. He served as Co-Chair for the Robotics andAutomation Society Student Activities Committee from 2012 to 2014.

Albert Wang (S’14) received the B.S. and M.S.degrees from the Department of Mechanical En-gineering, Massachusetts Institute of Technology,Cambridge, MA, USA, in 2010 and 2012 respec-tively. He is currently a Ph.D. Candidate in theMIT Biomimetic Robotics Lab, working on theHERMES humanoid project. His research interestsinclude dynamic legged locomotion, actuation sys-tems, and system architecture, and shared autonomyfor teleoperation.

Sangok Seok (S’10, M’14) received the B.S. andM.S. degrees from the School of Mechanical andAerospace Engineering, Seoul National University,Seoul, Korea, in 2002 and 2004, respectively, and thePh.D. degree in Mechanical Engineering from theMassachusetts Institute of Technology, Cambridge,MA, USA, in 2014.

He was with the Korean branch of National In-struments as an applications engineer and a mar-keting engineer from 2004 to 2009, and with theMechatronics R&D center, Samsung Electronics as

a principal engineer from 2014 to 2015. He is currently a leader of the roboticsgroup at Naver Labs. His research focus is on service robot platforms, whichenable ambient intelligence, utilizing space and mobility. Dr. Seok receivedthe IEEE/ASME Transactions on Mechatronics best paper award for 2016.

David Otten received the B.S. and S.M. degreesfrom the Massachusetts Institute of Technology(MIT), Cambridge in 1973 and 1974 respectively.

In 1974 he joined the MIT Electric Power SystemsEngineering Laboratory (EPSEL) as a staff engineer.Since 1984 he has been a Principal Research Engi-neer in the renamed Laboratory for Electromagneticand Electronic System (LEES) at MIT. His researchinterests include instrumentation, power electronics,and the micromouse robot contest.

Jeffrey Lang (F’98) received his SB (1975), SM(1977) and PhD (1980) degrees from the Departmentof Electrical Engineering and Computer Scienceat the Massachusetts Institute of Technology. Hejoined the faculty of MIT in 1980 where he is nowthe Vitesse Professor of Electrical Engineering. Heserved as the Associate Director of the MIT Labo-ratory for Electromagnetic and Electronic Systemsfrom 1991 to 2003 and now serves as an AssociateDirector of the MIT Microsystems Technology Lab-oratories.

Professor Lang’s research and teaching interests focus on the analysis,design and control of electromechanical systems with an emphasis on: rotatingmachinery; micro/nano-scale (MEMS/NEMS) sensors, actuators and energyconverters; flexible structures; and the dual use of electromechanical actuatorsas motion and force sensors. He has written over 280 papers and holds26 patents in the areas of electromechanics, MEMS, power electronics andapplied control. He has been awarded 4 best-paper prizes from IEEE societies,and has received two teaching awards from MIT. He is a coauthor of”Foundations of Analog and Digital Electronic Circuits” published by MorganKaufman, and the editor of, and a contributor to, ”Multi-Wafer RotatingMEMS Machines: Turbines Generators and Engines” published by Springer.Professor Lang is a Fellow of the IEEE, and a former Hertz Foundation Fellow.He served as an Associate Editor of ”Sensors and Actuators” between 1991and 1994. He has also served as the General Co-Chair and Technical Co-Chairof the 1992 and 1993 IEEE MEMS Workshops, respectively, and the GeneralCo-Chair of the 2013 PowerMEMS Conference.

Sangbae Kim (M’08) received the B.S. degree inmechanical engineering from Yonsei University inKorea, in 2001 and earned the M.S and Ph.D. de-grees in the Biomimetics Laboratory, Stanford Uni-versity in 2004 and 2008 respectively. Prof. SangbaeKim is currently the director of the BiomimeticRobotics Laboratory and an Associate Professor ofMechanical Engineering at MIT.

His research focuses on bio-inspired robotic plat-form design by extracting principles from complexbiological systems. Kim’s achievements on bio-

inspired robot development include the world’s first directional adhesiveinspired from gecko lizards, and a climbing robot, Stickybot, that utilizesthe directional adhesives to climb smooth surfaces, featured in TIME’s bestinventions of 2006. The MIT Cheetah achieves stable outdoor running at anefficiency of animals, employing biomechanical principles from studies ofthe best runners in nature. This achievement was covered by more than 200articles. He was a recipient of the King-Sun Fu Memorial Best Transactionson Robotics Paper Award (2008), DARPA YFA (2013), and NSF CAREER(2014) award.